The trade was first reported by TSN.
Wild prospect Nick Leddy is going to Chicago, as well, in the trade.
Johnsson will be an unrestricted free agent July 1 and has a $4.85 million salary cap hit this season. Barker has two more years left on his deal after this season at a $3.08 million cap hit.
The Blackhawks need to shed some serious payroll in the offseason to accommodate new contract extensions that kick in for Jonathon Toews, Patrick Kane and Duncan Keith. This is a beginning -- they shaved off $3 million for next season.
Barker, 23, was in his fourth full season with the Blackhawks. Leddy, 19, was the Wild's first-round draft pick in 2009.
ESPN.com's Pierre LeBrun contributed to this report.